Joseph Hart, 54, was found guilty of raping the girl between 1991 and 1994 when she was aged between four and seven.
He was also convicted of sexually abusing another girl when she was also aged between four and seven.
The High Court in Glasgow heard Hart's young victim told an adult what had happened to her, but there was no corroboration.
Hart, from Glasgow, was finally brought to justice when his second victim went to the police and an investigation began.
Judge Bill Dunlop, QC, placed Hart on the Sex Offenders' Register and remanded him in custody.
He will be sentenced next month.